This is manifestly independent on the cutoff. It is also manifestly zero, because it is given by a series of scaleless integrals.
In the context of effective field theories where the method of region is used, such as NRGR of SCET, the overlap contribution \(R\)  is usually referred as the "zero-bin contribution" [1]}. One can obtain the full overlap either by expanding the high-energy integral \(I_{(II)}\)  around the low-energy limit, or the integrand of the low-energy integral \(I_{(I)}\)  around the high-energy limit. Since the overlap is obtained by expanding single-scale integrals, \(I_{(I)}\)  or \(I_{(II)}\)  it is given by scaleless integrals which vanish in dimensional regularization.
We will now apply these techniques in the context of NRGR.
